Alternating graphs

In this paper we generalize the concept of alternating knots to alternating graphs and show that every abstract graph has a spatial embedding that is alternating. We also prove that every spatial graph is a subgraph of an alt ernating graph. We define n-composition for spatial graphs and generalize t he results of Menasco on alternating knots to show that an alternating grap h is n-composite For, n = 0, 1, 2, 3 if and only if it is "obviously n-comp osite" in any alternating projection. Moreover, no closed incompressible pa irwise incompressible surface exists in the complement of an alternating gr aph. We then generalize results of Kauffman, Murasugi, and Thistlethwaite t o prove that the crossing number of an even-valent rigid-vertex alternating spatial graph is realized in every reduced alternating projection with no uncrossed cycles and, if the graph is not 2-composite, the crossing number is not realized in any non-alternating projection. We give examples showing that this result does not hold For graphs with vertices of odd valence or graphs with uncrossed cycles. (C) 1999 Academic Press.